Finite Group Databasebeta

← #6468182

Subgroups — #6468182

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#29148C212yesmaximal, complemented
4#29131C214yesmaximal, complemented
5#29111C212yesmaximal, complemented
6#29109C212yesmaximal
7#28726C212yesmaximal, complemented
8#27969C212yesmaximal, complemented
9#25864C212yesmaximal
10#23817C214yesmaximal, complemented
11C4×#489C214yesmaximal, complemented
12#20734C212yesmaximal, complemented
13#20732C212yesmaximal
14#20463C211yesmaximal, characteristic, complemented
15C2×#533—21no—
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#489C2214yes—
20C2×#431C2212yes—
21#2868C2211yescharacteristic, complemented
22#2868C2211yescharacteristic, complemented
23#2862—21no—
24#2859—21no—
25#2859C2212yescomplemented
26#2857—21no—
27#2857—21no—
28#2857C2212yes—
29#2857C2212yes—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2212yescomplemented
33C4×C4⋊Q8C2212yescomplemented
34C4×#124—21no—
35C4×#124—21no—
36C4×#124—21no—
37C4×#124—21no—
38C4×#124C2212yes—
39C4×#124C2212yes—
40C4×#124C2212yes—
41C4×#124C2212yes—
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46#2285—21no—
47#2285C2212yes—
48C2×#382C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50C2×C4×C22⋊C4—21no—
51C2×C43C2211yescharacteristic
52#2101C2218yescomplemented
53#2072C2214yescomplemented
54#2071C2214yes—
55#2042C2218yes—
56#2024C2214yescomplemented
57#2023C2214yes—
58#1993C2214yescomplemented
59#1959C2214yes—
60#1936C2218yes—
61#1935C2218yescomplemented
62#1836C2214yescomplemented
63#1824C2214yes—
64#1823C2214yes—
65#1803C2214yescomplemented
66#1803C2214yescomplemented
67#1802C2214yes—
68#1802C2214yes—
69#1787C2214yes—
70#1786C2214yes—
71#1767C2212yescomplemented
72#1767C2212yescomplemented
73C4×#103C2218yescomplemented
74C4×#102C2218yes—
75#1749C2212yes—
76#1749C2212yes—
77#1744C2212yescomplemented
78C4×#97C2212yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82#533—48no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×#124—28no—
91C2×#124—21no—
92C2×#124—21no—
93C2×#124C2312yes—
94C2×#124C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—28no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—28no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#489—216no—
127#431—28no—
128C2×C4⋊C8C2314yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135C2×#97C2314yes—
136C2×C4×C8C2314yes—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—21no—
143#391C2312yes—
144#391C2312yes—
145#390—21no—
146#390—21no—
147#390C2312yes—
148#390C2312yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384—21no—
169#384C2312yes—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#382—22no—
174#382—22no—
175#382—21no—
176#382—21no—
177#382C2312yes—
178#382C2312yes—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C25—21no—
214C22×Q8—24no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—24no—
219C22×Q8—42no—
220C22×Q8C2412yes—
221C22×D4—24no—
222C22×D4—21no—
223C22×D4—24no—
224C22×D4—42no—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8C2418yes—
242C4⋊Q8—28no—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248#125—48no—
249#124—28no—
250#124—28no—
251#124—24no—
252#124—24no—
253#124—24no—
254#124#4118yes—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257C4×Q8—28no—
258C4×Q8—28no—
259C4×Q8—48no—
260C4×Q8—48no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—44no—
266C4×D4—44no—
267#117—48no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2414yes—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42#4118yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C4⋊C8—28no—
347#103—216no—
348#103—48no—
349#102—216no—
350#102—48no—
351C22⋊C8—28no—
352#97—28no—
353#97—44no—
354C4×C8—28no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—416no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8C2×#4114yes—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—416no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C2×#4114yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—816no—
435C22×C4#12414yes—
436C22×C4#12414yes—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4#13714yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4114yes—
453C22×C4C2511yescharacteristic
454C2×C8—28no—
455C2×C8—28no—
456C2×C8—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—44no—
512C42—44no—
513C42—48no—
514C42#12414yes—
515C42#12414yes—
516C42C2×#4114yes—
517C42C2×#4114yes—
518C42C2×#4114yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23C2×#12412yes—
538C23C2×AGL(1,ℤ/8)12yes—
539C23C2×#13712yes—
540C23C23×D411yescharacteristic
541Q8—28no—
542Q8—48no—
543Q8—44no—
544Q8—88no—
545Q8—88no—
546D4—28no—
547D4—48no—
548D4—44no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4—88no—
601C2×C4#489116yescomplemented
602C2×C4#52512yes—
603C2×C4#52512yes—
604C2×C4C2×#12412yes—
605C2×C4C2×#12412yes—
606C2×C4C2×#12412yes—
607C2×C4C2×#12412yes—
608C2×C4C2×#12412yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4#57512yes—
614C2×C4#57512yes—
615C2×C4C22×#4112yes—
616C2×C4C22×#4112yes—
617C2×C4#58512yes—
618C8—48no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—816no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×#48914yes—
634C22C2×#52511yescharacteristic
635C22C22×#12411yescharacteristic
636C22#344911yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C22×#13711yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57511yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—44no—
651C4—41no—
652C4—41no—
653C4—84no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4C2×#48914yes—
658C4C2×#48914yes—
659C4#347514yes—
660C2—24no—
661C2—84no—
662C2#3494611yescharacteristic
663C2#6056111yescharacteristic
664C2#6056211yescharacteristic
665C2C22×#48911yescharacteristic
666C2#6061811yescharacteristic
667C2#6094911yescharacteristic
668C2#6095011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36645C2semidirect
3. #29148C2semidirect
4. #29131C2semidirect
5. #29111C2semidirect
7. #28726C2semidirect
8. #27969C2semidirect
10. #23817C2semidirect
11. C4×#489C2semidirect
12. #20734C2semidirect
14. #20463C2semidirect
21. #2868C22semidirect
22. #2868C22semidirect
25. #2859C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×C4⋊Q8C22semidirect
52. #2101C22semidirect
53. #2072C22semidirect
56. #2024C22semidirect
58. #1993C22semidirect
61. #1935C22semidirect
62. #1836C22semidirect
65. #1803C22semidirect
66. #1803C22semidirect
71. #1767C22semidirect
72. #1767C22semidirect
73. C4×#103C22semidirect
77. #1744C22semidirect
78. C4×#97C22semidirect
601. C2×C4#489semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#366452
C2×C4×#1242
#291482
#291312
#291112
#291092
#287262
#279692
#258642
#238172
C4×#4892
#207342
#207322
#204632